#115446

order	512
nilpotency class	3
isoclinism	isoclinic class of #115437
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

924 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,5,3)(4,6,8,7)(9,10,14,19,15,20,13,18)(11,16,22,30,23,31,28,34)(12,24,27,17,25,33,26,32)(21,35,37,40,36,39,38,29);(2,6)(3,7)(4,8)(9,11)(10,16)(12,21)(13,28)(14,22)(15,23)(17,29)(18,34)(19,30)(20,31)(24,39)(25,36)(26,38)(27,37)(32,40)(33,35);(1,3,5,2)(4,6,8,7)(9,12)(10,17)(11,21)(13,27)(14,26)(15,25)(16,29)(18,33)(19,24)(20,32)(22,38)(23,36)(28,37)(30,39)(31,40)(34,35)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22642);